The Hardness of Approximating the Threshold Dimension, Boxicity and Cubicity of a Graph

The threshold dimension of a graph G(V,E) is the smallest integer k such that E can be covered by k threshold spanning subgraphs of G. A k-dimensional box is the Cartesian product R1 ×R2 × · · · × Rk where each Ri is a closed interval on the real line. The boxicity of a graph G, denoted as box(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-dimensional boxes. A unit cube in k-dimensional space or a k-cube is defined as the Cartesian product R1 ×R2 × · · · ×Rk where each Ri is a closed interval on the real line of the form [ai, ai + 1]. The cubicity of G, denoted as cub(G), is the minimum integer k such that G can be represented as the intersection graph of a collection of k-cubes. In this paper we will show that there exists no polynomial-time algorithm to approximate the threshold dimension of a graph on n vertices with a factor of O(n) for any ǫ > 0, unless NP = ZPP . From this result we will show that there exists no polynomial-time algorithm to approximate the boxicity and the cubicity of a graph on n vertices with factor O(n) for any ǫ > 0, unless NP = ZPP . In fact all these hardness results hold even for a highly structured class of graphs namely the split graphs. We will also show that it is NP-complete to determine if a given split graph has boxicity at most 3.